例題ではじめる部分空間法 - パターン認識へのいざない -
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- ほだか かたづ
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1 - - ( ) (1)
2 ( ) MATLAB/Octave 3 download s-hotta/rsj2012 (2)
3 ( ) [1] 対応付け 未知パターン ( クラスが未知 ) 利用 クラス ( 概念 ) 9 訓練パターン ( クラスが既知 ) (3)
4 [1] 識別演算部 未知パターン 前処理部 特徴抽出部 照合識別辞書 5 出力 識別部 (4)
5 x 2 0 x x 1 x 1 ( ) (5)
6 (Bayes decision rule) max j P (ω j x) = P (ω k x) x ω k P (ω j x) = P (ω j)p(x ω j ) j P (ω j)p(x ω j ) P (ω j ): ω j ( ) P (ω j x): x ω j ( ) p(x ω j ): ω j x ( ) p(x ω j ) ( ) (6)
7 [2, 3, 4] : ( ) ( ) (7)
8 : ( ) : ( ) 学習 クラスごとに訓練標本を部分空間で圧縮 エントロピー最小化 顔 5 猫 バイク 認識 未知標本 ~ c j 1 + c j 2 + c jr c + c k1 k 2 + ckr = = 認識結果顔 バイクで顔を表現できない (8)
9 : min u i d p(u i ) log p(u i ) i=1 : + r i=1 w i(u i x) 2 x x u u 2 1 T U j x x (9)
10 (1925- ) OCR ASPET/71 & ( ) ( ) ( ) ( ) (10)
11 1/3 [5] 0 subspace sub () ( super) ( ) ( ) (11)
12 2/ (12)
13 3/3 線型部分空間でなければ同じ直線上に存在する点同士の足し算 引き算 スカラー倍が同じ直線上にのらない (affine subspace, linear manifold, linear variety) (13)
14 d : x = x 1.. x d, x = (x 1,..., x d ) x x = d i=1 x2 i = x 2 x : x = x x : cos θ = x y x y : x y 2 = 2(1 cos θ) x y 半径 1 n + 1 n (n 3 ) (14)
15 r d u 1, u 2,..., u r d r ( ) U = (u 1 u 2 u r ) (U U = I) x c 2 u 2 u 1 0 x~ c 1 R r x c = U x = (u 1 x u 2 x u r x) R d x x = Uc = UU x (x U ) UU (15)
16 x x i x i ( ) x = x 1 x d (1) f = x x = d i=1 x2 i x ( ) x f = f = f x 1 f x d = (2x1 2x d ) = 2x (2) d d A f = x Ax x x f = 2Ax (3) f = y Ax x x f = A y (16)
17 arg max subject to arg max subject to max f(x) f(x) max f(x) x =1 x = 1 x f(x) argmax x f(x) f(x) x arg argmin f(x) x subject to s.t. max f(x), s.t. x x f(x) (17)
18 [6] SVM g(x) = 0 f(x) λ L = f(x) λg(x) 1 x L = f λ g = 0, L λ = 0 d + 1 x 1,...,x d, λ d (18)
19 ( ) A u u = 1 u Au max u Au u s.t. u u 1 = 0 λ L = u Au λ(u u 1) L/ u = 0, L/ λ = 0 L/ u = 2Au 2λu = 0 Au = λu 2 A ( u u u = 1 ) 2 2 (19)
20 (subspace method) [2, 4] [7] [8] ( ) (20)
21 n d x i (i = 1,..., n) C j U j x x max j=1,...,c { U j x } = U k x x class k (21)
22 class j class j x U T j d j x x U j 0 x 部分空間法 0 x 最小距離法 x d 2 j = x 2 U j x 2 x 2 U j x 2 (22)
23 cos θ 1/3 U d r 1 x U Uc c c = 1 : max x Uc = cos θ c s.t. c c 1 = 0 (1) x = 1 Uc = (Uc) (Uc) = c Ic = 1 j x u u 2 1 Uc (23)
24 cos θ 2/3 (1) L = x Uc λ 2 (c c 1) L/ c = U x λc = 0 c = 1 λ U x (2) c c = 1 λ 2 (U x) (U x) = 1 λ 2 U x 2 = 1 λ = U x λ (2) c = U x U x x U 1 (24)
25 cos θ 3/3 c = (U x)/ U x x Uc cos θ = x Uc = x UU x U x = U x 2 U x = U x = λ cos θ (U x)/ U x R r (UU x)/ U x R d cos θ U x x u u 2 1 T UU x Uc = T U x (25)
26 U 1/2 0 (26)
27 U 2/2 n x 1,..., x n d i = x i 2 (u 1 x i ) 2 u 1 x i 2 u 1 (u 1 x i ) 2 u 1 : ( n n ) max (u 1 x i ) 2 = u 1 x i x i u 1 u 1 i=1 i=1 s.t. u 1 u 1 1 = 0 D = n i=1 x ix i λ 1 r u r D r λ r r U j = (u 1 u r ) (27)
28 d d n X = (x 1 x n ) d d d D = XX n d n n N = X X U X X n ( n ) (28)
29 N = X X R n n r 0 λ 1,..., λ r (D = XX ) v 1,..., v r R n i d u i [9] u i = ±Xv i λi ( EVD.m ) (29)
30 : : [10, 11] P (ω j x) = P (ω j)p(x ω j ) j P (ω j)p(x ω j ) (30)
31 ω j N (x µ j, Σ j ) = 1 (2π) d 2 Σ j 1 2 { exp 1 } 2 (x µ j) Σ 1 j (x µ j ) µ j : d 1 Σ j : d d x i x i (31)
32 () (32)
33 ˆµ j = 0, ˆΣj = R j R j = 1 n j nj i=1 x ix i : N (x ˆµ j, ˆΣ j ) = ( 1 exp 1 ) (2π) d/2 R j 1/2 2 x R 1 j x (33)
34 N (x ˆµ j, ˆΣ j ) = ( 1 (2π) d/2 exp 1 ) R j 1/2 2 x R 1 j x g j (x) = Λ 1 2 j U j x 2 d i=1 ln λ ji R j = U j Λ j U j (34)
35 g j (x) = Λ 1 2 j U j x 2 d i=1 ln λ ji x i x i ( ) (35)
36 w 1 w 2 w d > 0 g j (x) = W 1 2 U j x 2 = d i=1 1 w i (u ji x)2 x x x 2 ( u T x) ji 2 0 u ji x u T ji (36)
37 1/w 1 1/w 2 1/w d > 0 w 1 w 2 w d > 0 ( ) w i : S j (x) = r i=1 w i(u ji x)2 = W 1 2 U j x 2 = w c j : S j (x) x ( ) (37)
38 CLAFIC ( ): w i = 1 (i = 1,..., r) ( ): w i = λ ji/λ j1 : w i = r i + 1 (i = 1,..., r) CLAFIC weight value proposed multiple similarity dimensionality r (38)
39 r S j (x) = (r i + 1)(u jix) 2 i=1 = r(u j1x) 2 + (r 1)(u j2x) (u jrx) 2 r r 1 = (u jix) 2 + (u jix) (u j1x) 2 i=1 i=1 (39)
40 1 USPS makedata.m WSC.m (40)
41 makedata.m makedata.m USPS [ 1, +1] 0 pair-wise makedata.m [0, +1] 0 9 usps.mat (41)
42 WSC.m WSC.m WSC.m Figure 1 10 Figure 2 imgnum U j (r = 13) 1 r imgnum (42)
43 : i j ij class total total (43)
44 C nclass d d n ndata i x i trai(:,ii) i trai_label(ii) i x test(:,ii) i test_label(ii) j U j U j C(j).U x y x y x *y x x = x x norm(x) (44)
45 Figure 1 (45)
46 Figure 2 test sample class 0 class 1 class 2 class 3 class 4 class 5 class 6 class 7 class 8 class 9 U j (U j U j x) (46)
47 λ = r i=1 λ i/ rank(d) j=1 λ j r ( ) 95 test validation accuracy [%] dimensionality of each subspace r (47)
48 r 95 linear weight ] [% c y ra 90 c u a CLAFIC 85 multiple similarity dimensionality r (48)
49 CPU 1.86GHz 2GB 32bit Windows MATLAB (R14) (r = 13) () SVM SVM () CLAFIC CLAFIC one-against-all one-against-all, RBF Kernel (49)
50 (%) (s) (KB) (r = 13) ( λ = 0.95) SVM SVM (50)
51 単体のパターンを観測しても何であるかはわからない 犬 同じクラスから由来する複数のパターン (CSM): (MSM): (51)
52 Compound Subspace Method (CSM) [11] P (ω X) = P (ω)p(x ω) ω P (ω)p(x ω) n T r k=1 f=1 i=1 f w i ( f u i f x k ) 2 Multiple Kernel Learning (52)
53 (compound Bayesian decision problem) [12] P (ω X) = P (ω)p(x ω) ω P (ω)p(x ω) n : X = (x 1 x 2 x n ) n (context): ω = (ω(1),..., ω(n)) x i ω(i) ひらめカレイカレイひらめ c (53)
54 (compound Bayesian decision problem) P (ω X) = P (ω)p(x ω) ω P (ω)p(x ω) ω c n p(x ω) CSM x i T (54)
55 CSM n x 1,..., x n i.i.d. P (ω j X ) = n P (ω j ) p(x i ω j ) i=1 c n P (ω j ) p(x i ω j ) j=1 i=1 n r n n S j(x ) = w l (u jlx i) 2 = W 1 2 U j x i 2 = S j(x i) i=1 l=1 i=1 i=1 (55)
56 CSM x T P (ω j F) = T P (ω j ) p f ( f x ω j ) f=1 c P (ω j ) j=1 f=1 T p f ( f x ω j ) (f ) F j (F) def = T r f f=1 l=1 T T f w l ( f u jl f x) 2 = f W 1 2 f U j f x 2 = f S j ( f x) f=1 f=1 (56)
57 CSM n X T P (ω j X ) = T n P (ω j ) p f ( f x i ω j ) c P (ω j ) f=1 i=1 T j=1 f=1 i=1 n p f ( f x i ω j ) C j ( X ) def = T f=1 i=1 n f S j ( f x i ) (57)
58 (mutual subspace method, MSM) [13] (cos θ ) (58)
59 V R d r d : r d < d U R d r i : r i < d Vb Uc max b,c (Vb) (Uc) = cos θ s.t. b b 1 = 0, c c 1 = 0 (3) (59)
60 1/2 (3) L = (Vb) (Uc) λ 2 (b b 1) µ 2 (c c 1) L/ b = 0, L/ c = 0 V Uc = λb (4) U Vb = µc (5) (4) c (5) b V UU Vb = λµb, U VV Uc = λµc (6) (60)
61 2/2 (4) b (5) c λ = µ b V Uc = b λb = λ (7) c U Vb = c µc = µ (8) (6) λ(= µ) V UU V (= U VV U ) cos θ V UU V U VV U (61)
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62 2 ETH-80 dataset [14] ( ) (30 ) (62)
63 2 dog1 dog png dog png dog png dog png dog png dog png dog png dog png dog png dog png dog png dog png dog png dog png dog png dog png dog png dog png dog png dog png dog png dog png dog png dog png dog png dog png dog png dog png dog png dog png dog png dog png dog png dog png do g png dog png dog png dog png dog png dog png dog png (63)
64 2 dog png (64)
65 make_data.m Y = R G B (65)
66 21 ( ) 10 ( ) 22 ( ) (1 11 ) (66)
67 (doubutsu.mat) IMG(:) x= IMG reshape(x, [3 3]) IMG(:): reshape: (67)
68 CSM.m MSM.m ri rd nclass ntrai ntest trai(ii).x test(ii).x EVD trai(ii).u test(ii).u trai(ii).label test(ii).label CONF ii ( ) ii ( ) ii ii ii ii (68)
69 CPU 2.8GHz 3.5GB 32bit Windows MATLAB (R14) WSC: MSM: CSM: (%) (s) WSC (r d = 20) MSM (r i = 5,r d = 6) CSM (r d = 20) (69)
70 r r ( ) tangent distance k-subspace clustering (fuzzy) k-varieties clustering (70)
71 ( d = 2) local subspace classifier ( ) subspace/ (71)
72 I [1],,,,,, Aug [2] S. Watanabe, P.F. Lambert, C.A. Kulikowski, J.L. Buxton, and R. Walker, Evaluation and selection of variables in pattern recognition, Comp. & Info. Sciences, vol. 2, (Julius Tou, ed.). New York: Academic Press, pp , [3] T. Iijima, H. Genchi, and K. Mori, A theory of character recognition by pattern matching method, Proc. of 1st Int l J. Conf. on Pattern Recognition, pp , [4] E. Oja, Subspace methods of pattern recognition, Research Studies Press, 1983., [5], 1, [6], - -,, [7] S. Watanabe, Knowing and guessing : A quantitative study of inference and information, John Wiley & Sons, New York, 1969.,, :,,, (72)
73 II [8] [9], - -,, [10],,, PRMU2010, vol. 110, no. 296, pp , Nov [11],,, PRMU2010, vol. 110, no. 330, pp , Dec [12] J.F. Hannan and H. Robbins, Asumptotic solutions of the compound decision problem for two completely specified distributions, Annals of Mathematical Statistcs, vol. 26, no. 1, pp , [13] (D), vol. J68-D, no. 3, pp , [14] B. Leibe and B. Schiele, Analyzing appearance and contour based methods for object categorization, Proc. of CVPR, pp , (73)
74 1 1: f ( x) = const. f g(x) = 0 g g(x) = 0 f(x) f = λ g (74)
75 2 2: Au = λu (u 0) u A λ x x ( ) 1/5 1/4 A A = x Ax 1/4 1/5 ( ) (75)
On the Limited Sample Effect of the Optimum Classifier by Bayesian Approach he Case of Independent Sample Size for Each Class Xuexian HA, etsushi WAKA
Journal Article / 学術雑誌論文 ベイズアプローチによる最適識別系の有限 標本効果に関する考察 : 学習標本の大きさ がクラス間で異なる場合 (< 論文小特集 > パ ターン認識のための学習 : 基礎と応用 On the limited sample effect of bayesian approach : the case of each class 韓, 雪仙 ; 若林, 哲史
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